Simply-Connected Spaces and Cohomology

Indeed, say that is simply-connected, so that any closed curve can be written as the boundary of some surface . Then we take any closed -form with . Stokes’ theorem tells us that

for any closed curve . But this means that every closed -form is path-independent, and path-independent -forms are exact. And so we conclude that , as asserted.

It will (eventually) turn out that the fact that both and vanish together is not a coincidence, but is in fact an example of a much deeper correspondence between homology and cohomology — between topology and analysis.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.